Definition:Unique up to Unique Morphism

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Definition

Let $\mathbf C$ be a category.

Let $S \subseteq \map {\operatorname {Ob} } {\mathbf C}$ be a subclass of its objects.


Definition 1

The class $S$ is unique up to unique morphism if and only if for all objects $s, t \in S$ there is a unique morphism from $s$ to $t$.


Definition 2

The class $S$ is unique up to unique morphism if and only if for all objects $s, t \in S$ there is a unique morphism from $s$ to $t$, and it is an isomorphism.


Also see


Weaker properties


Sources